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On the order of the Maass \(L\)-function on the critical line. (English) Zbl 0724.11029

Number theory. Vol. I. Elementary and analytic, Proc. Conf., Budapest/Hung. 1987, Colloq. Math. Soc. János Bolyai 51, 325-354 (1990).
Let \(\Psi\) be a Maass form for the full modular group with eigenvalue \(1/4+\kappa^2\). Suppose that \(\Psi\) is a Hecke eigenform, normalized by having \(\sqrt{y}K_{i\kappa}(2\pi y)e^{2\pi ix}\) as its Fourier term of order 1. Its \(L\)-function \(L_{\Psi}\) is shown to satisfy \[ L_{\Psi}(1/2+it) \ll_{\varepsilon} \| \Psi \| \sqrt{\kappa} e^{\pi \kappa /2} t^{1/3+\varepsilon} \] for \(t\gg 1\), \(\kappa \ll t^{1/3}\) and \(\varepsilon >0\), the \(L^2\)-norm of \(\Psi\) is denoted by \(\| \Psi \|\). This result improves the estimate \(O(t^{1/2+\varepsilon})\) in [C. Epstein, J. L. Hafner and P. Sarnak, Math. Z. 190, 113–128 (1985; Zbl 0565.10026)]. The proof is considerably more complicated.
The central point in the proof is the estimation of \(\int^{2N}_{M=N}| \sum^{2M}_{n\geq M}\rho (n)n^{-s}| \,dM;\) the \(\rho(n)\) are Fourier coefficients of \(\Psi\). The sum over \(n\) is split up into many small subsums. The subsums are treated by a method of M. Jutila [Lect. Notes Math. 1380, 120–136 (1989; Zbl 0674.10032)].
[For the entire collection see Zbl 0694.00005.]

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F37 Forms of half-integer weight; nonholomorphic modular forms
11M41 Other Dirichlet series and zeta functions